Probability & Statistics

A self-contained educational project illustrating core Probability and Statistics concepts through minimal, runnable Python code and SQL analytics. Designed so a practitioner can read the source, understand the algorithm, and immediately apply it to real data.

Concepts Covered

Descriptive Statistics (src/descriptive_stats/)

• Central tendency: arithmetic mean, median, mode, trimmed mean, geometric mean, harmonic mean
• Spread: variance, standard deviation, IQR, MAD, coefficient of variation
• Shape: skewness (Pearson), excess kurtosis
• Percentiles: P10, P25, P50, P75, P90, P95, P99
• Distribution fitting: Normal, Log-Normal, Gamma, Exponential, Beta — ranked by AIC
• Goodness-of-fit: Kolmogorov-Smirnov test
• Visualisations: histograms with fitted PDFs, QQ plots, empirical vs. theoretical CDFs

Hypothesis Testing (src/hypothesis_testing/)

• One-sample t-test: compare sample mean to known value
• Two-sample Welch's t-test: compare two groups without equal-variance assumption
• Chi-squared test of independence: categorical association
• Mann-Whitney U test: non-parametric two-sample test
• Two-proportion z-test: A/B testing for conversion rates
• Bootstrap confidence intervals: percentile method, B=10,000 iterations
• Power analysis: sample size calculation for t-test and z-test
• Multiple testing correction: Bonferroni (FWER) and Benjamini-Hochberg (FDR)
• Effect sizes: Cohen's d, Cohen's h, Cramér's V, rank-biserial correlation

Bayesian Analysis (src/bayesian_analysis/)

• Beta-Binomial model: conjugate prior for conversion rate A/B testing
• Bayesian updating: sequential posterior updates with each observation
• Monte Carlo sampling: P(B > A) and lift distribution with 100,000 samples
• Expected loss: risk-minimisation decision criterion
• Credible intervals: vs. frequentist confidence intervals (coverage comparison)
• Prior sensitivity: effect of vague vs. informative priors
• Normal-Normal and Gamma-Poisson conjugacy: for continuous and count data

SQL Analytics (src/sql_analytics/)

• Descriptive stats SQL: PERCENTILE_CONT, STDDEV_SAMP, VAR_SAMP, SKEWNESS, KURTOSIS
• Window functions: PERCENT_RANK, CUME_DIST, NTILE, RANK, DENSE_RANK
• Z-score normalisation: in SQL with outlier flagging
• IQR outlier detection: Tukey's fences entirely in SQL
• Histogram buckets: WIDTH_BUCKET for equal-width binning
• Grouped statistics: per-category descriptive stats
• Funnel analysis: step-by-step conversion and drop-off rates
• Cohort retention: weekly retention matrix
• Linear regression: REGR_SLOPE, REGR_INTERCEPT, REGR_R2, CORR

Project Structure

probability_statistics/
    src/
        notebooks/
            statistics_demo.ipynb       # Full workflow demo
        descriptive_stats/
            statistics_calculator.py    # Mean, median, spread, shape
            distribution_analyzer.py    # Distribution fitting and visualisation
        hypothesis_testing/
            test_runner.py              # t-tests, chi-squared, z-test, bootstrap, MTC
            power_analysis.py           # Sample size and power curves
        bayesian_analysis/
            bayesian_ab_test.py         # Beta-Binomial A/B testing
            conjugate_priors.py         # Conjugate prior families
        sql_analytics/
            sql_statistics.py           # DuckDB SQL statistical queries
    docs/
        concepts.md                     # Theory + mermaid diagrams
        flow_diagrams.md                # Pipeline flow diagrams
    docker/
        Dockerfile
        docker-compose.yml
        requirements.txt
    outputs/                            # Generated plots (auto-created)
    logs/                               # Log files (auto-created)
    config.yaml                         # All constants and hyperparameters
    pyproject.toml
    README.md

Usage

All configuration is in config.yaml. No command-line arguments.

Run individual modules directly

# From the project root (probability_statistics/)

# Descriptive statistics
python src/descriptive_stats/statistics_calculator.py
python src/descriptive_stats/distribution_analyzer.py

# Hypothesis testing
python src/hypothesis_testing/test_runner.py
python src/hypothesis_testing/power_analysis.py

# Bayesian analysis
python src/bayesian_analysis/bayesian_ab_test.py
python src/bayesian_analysis/conjugate_priors.py

# SQL analytics (DuckDB)
python src/sql_analytics/sql_statistics.py

Run via Jupyter notebook

cd src/notebooks
jupyter notebook statistics_demo.ipynb

Run via Docker

cd docker

# Build image
docker-compose build

# Run SQL analytics demo
docker-compose run probability_statistics

# Run descriptive stats
docker-compose run descriptive_stats

# Run hypothesis testing
docker-compose run hypothesis_testing

# Run Bayesian A/B test
docker-compose run bayesian_ab

# Start Jupyter server (access at http://localhost:8888)
docker-compose up jupyter

Configuration (config.yaml)

All hyperparameters and constants live in config.yaml. Key sections:

Outputs

All plots are saved to outputs/ as PNG files:

Key References

• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.)
• Gelman, A., et al. (2013). Bayesian Data Analysis (3rd ed.)
• VanderPlas, J. (2016). Python Data Science Handbook (NumPy, SciPy, Pandas)
• Kohavi, R., et al. (2020). Trustworthy Online Controlled Experiments (A/B testing)
• DuckDB Documentation: https://duckdb.org/docs/sql/functions/aggregates